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Data evaluation and data combination require the use of a wide
range of probability theory concepts and tools, from deductive
statistics mainly concerning frequencies and sample tallies to
inductive inference for assimilating non-frequency data and a
priori knowledge. Computational Methods for Data Evaluation and
Assimilation presents interdisciplinary methods for integrating
experimental and computational information. This self-contained
book shows how the methods can be applied in many scientific and
engineering areas. After presenting the fundamentals underlying the
evaluation of experimental data, the book explains how to estimate
covariances and confidence intervals from experimental data. It
then describes algorithms for both unconstrained and constrained
minimization of large-scale systems, such as time-dependent
variational data assimilation in weather prediction and similar
applications in the geophysical sciences. The book also discusses
several basic principles of four-dimensional variational
assimilation (4D VAR) and highlights specific difficulties in
applying 4D VAR to large-scale operational numerical weather
prediction models.
Data evaluation and data combination require the use of a wide
range of probability theory concepts and tools, from deductive
statistics mainly concerning frequencies and sample tallies to
inductive inference for assimilating non-frequency data and a
priori knowledge. Computational Methods for Data Evaluation and
Assimilation presents interdisciplinary methods for integrating
experimental and computational information. This self-contained
book shows how the methods can be applied in many scientific and
engineering areas. After presenting the fundamentals underlying the
evaluation of experimental data, the book explains how to estimate
covariances and confidence intervals from experimental data. It
then describes algorithms for both unconstrained and constrained
minimization of large-scale systems, such as time-dependent
variational data assimilation in weather prediction and similar
applications in the geophysical sciences. The book also discusses
several basic principles of four-dimensional variational
assimilation (4D VAR) and highlights specific difficulties in
applying 4D VAR to large-scale operational numerical weather
prediction models.
This book addresses the experimental calibration of best-estimate
numerical simulation models. The results of measurements and
computations are never exact. Therefore, knowing only the nominal
values of experimentally measured or computed quantities is
insufficient for applications, particularly since the respective
experimental and computed nominal values seldom coincide. In the
author's view, the objective of predictive modeling is to extract
"best estimate" values for model parameters and predicted results,
together with "best estimate" uncertainties for these parameters
and results. To achieve this goal, predictive modeling combines
imprecisely known experimental and computational data, which calls
for reasoning on the basis of incomplete, error-rich, and
occasionally discrepant information. The customary methods used for
data assimilation combine experimental and computational
information by minimizing an a priori, user-chosen, "cost
functional" (usually a quadratic functional that represents the
weighted errors between measured and computed responses). In
contrast to these user-influenced methods, the BERRU (Best Estimate
Results with Reduced Uncertainties) Predictive Modeling methodology
developed by the author relies on the thermodynamics-based maximum
entropy principle to eliminate the need for relying on minimizing
user-chosen functionals, thus generalizing the "data adjustment"
and/or the "4D-VAR" data assimilation procedures used in the
geophysical sciences. The BERRU predictive modeling methodology
also provides a "model validation metric" which quantifies the
consistency (agreement/disagreement) between measurements and
computations. This "model validation metric" (or "consistency
indicator") is constructed from parameter covariance matrices,
response covariance matrices (measured and computed), and response
sensitivities to model parameters. Traditional methods for
computing response sensitivities are hampered by the "curse of
dimensionality," which makes them impractical for applications to
large-scale systems that involve many imprecisely known parameters.
Reducing the computational effort required for precisely
calculating the response sensitivities is paramount, and the
comprehensive adjoint sensitivity analysis methodology developed by
the author shows great promise in this regard, as shown in this
book. After discarding inconsistent data (if any) using the
consistency indicator, the BERRU predictive modeling methodology
provides best-estimate values for predicted parameters and
responses along with best-estimate reduced uncertainties (i.e.,
smaller predicted standard deviations) for the predicted
quantities. Applying the BERRU methodology yields optimal,
experimentally validated, "best estimate" predictive modeling tools
for designing new technologies and facilities, while also improving
on existing ones.
The computational models of physical systems comprise parameters,
independent and dependent variables. Since the physical processes
themselves are seldom known precisely and since most of the model
parameters stem from experimental procedures which are also subject
to imprecisions, the results predicted by these models are also
imprecise, being affected by the uncertainties underlying the
respective model. The functional derivatives (also called
“sensitivities”) of results (also called “responses”)
produced by mathematical/computational models are needed for many
purposes, including: (i) understanding the model by ranking the
importance of the various model parameters; (ii) performing
“reduced-order modeling” by eliminating unimportant parameters
and/or processes; (iii) quantifying the uncertainties induced in a
model response due to model parameter uncertainties; (iv)
performing “model validation,” by comparing computations to
experiments to address the question “does the model represent
reality?” (v) prioritizing improvements in the model; (vi)
performing data assimilation and model calibration as part of
forward “predictive modeling” to obtain best-estimate predicted
results with reduced predicted uncertainties; (vii) performing
inverse “predictive modeling”; (viii) designing and optimizing
the system. This 3-Volume monograph describes a comprehensive
adjoint sensitivity analysis methodology, developed by the author,
which enables the efficient and exact computation of arbitrarily
high-order sensitivities of model responses in large-scale systems
comprising many model parameters. The qualifier “comprehensive”
is employed to highlight that the model parameters considered
within the framework of this methodology also include the
system’s uncertain boundaries and internal interfaces in
phase-space. The model’s responses can be either scalar-valued
functionals of the model’s parameters and state variables (e.g.,
as customarily encountered in optimization problems) or general
function-valued responses. Since linear operators admit bona-fide
adjoint operators, responses of models that are linear in the state
functions (i.e., dependent variables) can depend simultaneously on
both the forward and the adjoint state functions. Hence, the
sensitivity analysis of such responses warrants the treatment of
linear systems in their own right, rather than treating them as
particular cases of nonlinear systems. This is in contradistinction
to responses for nonlinear systems, which can depend only on the
forward state functions, since nonlinear operators do not admit
bona-fide adjoint operators (only a linearized form of a nonlinear
operator may admit an adjoint operator). Thus, Volume 1 of this
book presents the mathematical framework of the nth-Order
Comprehensive Adjoint Sensitivity Analysis Methodology for
Response-Coupled Forward/Adjoint Linear Systems (abbreviated as
“nth-CASAM-L”), which is conceived for the most efficient
computation of exactly obtained mathematical expressions of
arbitrarily-high-order (nth-order) sensitivities of a generic
system response with respect to all of the parameters underlying
the respective forward/adjoint systems. Volume 2 of this book
presents the application of the nth-CASAM-L to perform a
fourth-order sensitivity and uncertainty analysis of an OECD/NEA
reactor physics benchmark which is representative of a large-scale
model comprises many (21,976) uncertain parameters, thereby amply
illustrating the unique potential of the nth-CASAM-L to enable the
exact and efficient computation of chosen high-order response
sensitivities to model parameters. Volume 3 of this book presents
the “nth-Order Comprehensive Adjoint Sensitivity Analysis
Methodology for Nonlinear Systems” (abbreviation: nth-CASAM-N)
for the practical, efficient, and exact computation of
arbitrarily-high order sensitivities of responses to model
parameters for systems that are also nonlinear in their underlying
state functions. Such computations are not feasible with any other
methodology. The application of the nth-CASAM-L and the nth-CASAM-N
overcomes the so-called “curse of dimensionality” in
sensitivity and uncertainty analysis, thus revolutionizing all of
the fields of activities which require accurate computation of
response sensitivities. Since this monograph includes many
illustrative, fully worked-out, paradigm problems, it can serve as
a textbook or as supplementary reading for graduate courses in
academic departments in the natural sciences and engineering.
The Second-Order Adjoint Sensitivity Analysis Methodology
generalizes the First-Order Theory presented in the author's
previous books published by CRC Press. This breakthrough has many
applications in sensitivity and uncertainty analysis, optimization,
data assimilation, model calibration, and reducing uncertainties in
model predictions. The book has many illustrative examples that
will help readers understand the complexity of the subject and will
enable them to apply this methodology to problems in their own
fields. Highlights: * Covers a wide range of needs, from graduate
students to advanced researchers * Provides a text positioned to be
the primary reference for high-order sensitivity and uncertainty
analysis * Applies to all fields involving numerical modeling,
optimization, quantification of sensitivities in direct and inverse
problems in the presence of uncertainties. About the Author: Dan
Gabriel Cacuci is a South Carolina SmartState Endowed Chair
Professor and the Director of the Center for Nuclear Science and
Energy, Department of Mechanical Engineering at the University of
South Carolina. He has a Ph.D. in Applied Physics, Mechanical and
Nuclear Engineering from Columbia University. He is also the
recipient of many awards including four honorary doctorates, the
Ernest Orlando Lawrence Memorial award from the U.S. Dept. of
Energy and the Arthur Holly Compton, Eugene P. Wigner and the Glenn
Seaborg Awards from the American Nuclear Society.
This text describes a comprehensive adjoint sensitivity analysis
methodology (nth-CASAM), developed by the author, which enablesthe
efficient and exact computation of arbitrarily high-order
functional derivatives of model responses to model parameters in
large-scale systems. The nth-CASAM framework is set in linearly
increasing Hilbert spaces, each of state-function-dimensionality,
as opposed to exponentially increasing parameter-dimensional
spaces, thereby overcoming the so-called “curse of
dimensionality” in sensitivity and uncertainty analysis. The
nth-CASAM is applicable to any model; the larger the number of
model parameters, the more efficient the nth-CASAM becomes for
computing arbitrarily high-order response sensitivities. The book
will be helpful to those working in the fields of sensitivity
analysis, uncertainty quantification, model validation,
optimization, data assimilation, model calibration, sensor fusion,
reduced-order modelling, inverse problems and predictive modelling.
This Volume Two, the second of three, presents the large-scale
application of the nth-CASAM to perform a representative
fourth-order sensitivity analysis of the Polyethylene-Reflected
Plutonium benchmark described in the Nuclear Energy Agency (NEA)
International Criticality Safety Benchmark Evaluation Project
(ICSBEP) Handbook. This benchmark is modeled mathematically by the
Boltzmann particle transport equation, involving 21,976
imprecisely-known parameters, the numerical solution of which
requires representative large-scale computations. The sensitivity
analysis presented in this volume is the most comprehensive ever
performed in the field of reactor physics and the results presented
in this book prove, perhaps counter-intuitively, that many of the
4th-order sensitivities are much larger than the corresponding
3rd-order ones, which are, in turn, much larger than the 2nd-order
ones, all of which are much larger than the 1st-order
sensitivities. Currently, the nth-CASAM is the only known
methodology which enables such large-scale computations of exactly
obtained expressions of arbitrarily-high-order response
sensitivities.
This text describes a comprehensive adjoint sensitivity analysis
methodology (C-ASAM), developed by the author, enabling the
efficient and exact computation of arbitrarily high-order
functional derivatives of model responses to model parameters in
large-scale systems. The model’s responses can be either
scalar-valued functionals of the model’s parameters and state
variables (as customarily encountered, e.g., in optimization
problems) or general function-valued responses, which are often of
interest but are currently not amenable to efficient sensitivity
analysis. The C-ASAM framework is set in linearly increasing
Hilbert spaces, each of state-function-dimensionality, as opposed
to exponentially increasing parameter-dimensional spaces, thereby
breaking the so-called “curse of dimensionality” in sensitivity
and uncertainty analysis. The C-ASAM applies to any model; the
larger the number of model parameters, the more efficient the
C-ASAM becomes for computing arbitrarily high-order response
sensitivities. The text includes illustrative paradigm problems
which are fully worked-out to enable the thorough understanding of
the C-ASAM’s principles and their practical application. The book
will be helpful to those working in the fields of sensitivity
analysis, uncertainty quantification, model validation,
optimization, data assimilation, model calibration, sensor fusion,
reduced-order modelling, inverse problems and predictive modelling.
It serves as a textbook or as supplementary reading for graduate
course on these topics, in academic departments in the natural,
biological, and physical sciences and engineering. This Volume
Three, the third of three, covers systems that are nonlinear in the
state variables, model parameters and associated responses. The
selected illustrative paradigm problems share these general
characteristics. A separate Volume One covers systems that are
linear in the state variables.
The computational models of physical systems comprise parameters,
independent and dependent variables. Since the physical processes
themselves are seldom known precisely and since most of the model
parameters stem from experimental procedures which are also subject
to imprecisions, the results predicted by these models are also
imprecise, being affected by the uncertainties underlying the
respective model. The functional derivatives (also called
"sensitivities") of results (also called "responses") produced by
mathematical/computational models are needed for many purposes,
including: (i) understanding the model by ranking the importance of
the various model parameters; (ii) performing "reduced-order
modeling" by eliminating unimportant parameters and/or processes;
(iii) quantifying the uncertainties induced in a model response due
to model parameter uncertainties; (iv) performing "model
validation," by comparing computations to experiments to address
the question "does the model represent reality?" (v) prioritizing
improvements in the model; (vi) performing data assimilation and
model calibration as part of forward "predictive modeling" to
obtain best-estimate predicted results with reduced predicted
uncertainties; (vii) performing inverse "predictive modeling";
(viii) designing and optimizing the system. This 3-Volume monograph
describes a comprehensive adjoint sensitivity analysis methodology,
developed by the author, which enables the efficient and exact
computation of arbitrarily high-order sensitivities of model
responses in large-scale systems comprising many model parameters.
The qualifier "comprehensive" is employed to highlight that the
model parameters considered within the framework of this
methodology also include the system's uncertain boundaries and
internal interfaces in phase-space. The model's responses can be
either scalar-valued functionals of the model's parameters and
state variables (e.g., as customarily encountered in optimization
problems) or general function-valued responses. Since linear
operators admit bona-fide adjoint operators, responses of models
that are linear in the state functions (i.e., dependent variables)
can depend simultaneously on both the forward and the adjoint state
functions. Hence, the sensitivity analysis of such responses
warrants the treatment of linear systems in their own right, rather
than treating them as particular cases of nonlinear systems. This
is in contradistinction to responses for nonlinear systems, which
can depend only on the forward state functions, since nonlinear
operators do not admit bona-fide adjoint operators (only a
linearized form of a nonlinear operator may admit an adjoint
operator). Thus, Volume 1 of this book presents the mathematical
framework of the nth-Order Comprehensive Adjoint Sensitivity
Analysis Methodology for Response-Coupled Forward/Adjoint Linear
Systems (abbreviated as "nth-CASAM-L"), which is conceived for the
most efficient computation of exactly obtained mathematical
expressions of arbitrarily-high-order (nth-order) sensitivities of
a generic system response with respect to all of the parameters
underlying the respective forward/adjoint systems. Volume 2 of this
book presents the application of the nth-CASAM-L to perform a
fourth-order sensitivity and uncertainty analysis of an OECD/NEA
reactor physics benchmark which is representative of a large-scale
model comprises many (21,976) uncertain parameters, thereby amply
illustrating the unique potential of the nth-CASAM-L to enable the
exact and efficient computation of chosen high-order response
sensitivities to model parameters. Volume 3 of this book presents
the "nth-Order Comprehensive Adjoint Sensitivity Analysis
Methodology for Nonlinear Systems" (abbreviation: nth-CASAM-N) for
the practical, efficient, and exact computation of arbitrarily-high
order sensitivities of responses to model parameters for systems
that are also nonlinear in their underlying state functions. Such
computations are not feasible with any other methodology. The
application of the nth-CASAM-L and the nth-CASAM-N overcomes the
so-called "curse of dimensionality" in sensitivity and uncertainty
analysis, thus revolutionizing all of the fields of activities
which require accurate computation of response sensitivities. Since
this monograph includes many illustrative, fully worked-out,
paradigm problems, it can serve as a textbook or as supplementary
reading for graduate courses in academic departments in the natural
sciences and engineering.
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